We have developed a novel experimental technique for tracking small particles in three dimensions with nanometer accuracy. The longitudinal positioning of a micrometer-sized particle is determined by using the Fresnel approximation to describe the transverse distribution of the wavefront that originated in the particle. The method utilizes the high-sensitivity quantitative phase imaging capability of diffraction phase microscopy recently developed in our laboratory. We demonstrate the principle of the technique with experiments on Brownian particles jittering in water both in bulk and in the vicinity of a boundary. The particles are localized in space within an error cube of 20 nm x 20 nm x 20 nm for a 33 Hz acquisition rate and 20s recording time.